How long will it take for a 18.0-g sample of iodine-131 to decay to leave a total of 2.25 g of the isotope? The half-life of iodine-131 is 8.07 days

k = 0.693/t1/2

Then ln(No/N) = kt
No = 18/-
N = 2.25
k from above.
Solve for t.

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To determine the time it takes for a sample of iodine-131 to decay to a certain amount, we can use the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the sample to decay.

In this case, the half-life of iodine-131 is given as 8.07 days. This means that after 8.07 days, half of the initial sample will have decayed, leaving us with 9.0 g of iodine-131.

To find out how many half-lives it takes for the 18.0 g sample to decay to 2.25 g, we can use the formula:

(Given Mass) = (Initial Mass) * (1/2)^(Number of Half-Lives)

Plugging in the values, we have:

2.25 g = 18.0 g * (1/2)^(Number of Half-Lives)

Next, we can isolate the number of half-lives by rearranging the equation:

(1/2)^(Number of Half-Lives) = 2.25 g / 18.0 g

Now, let's calculate this value:

(1/2)^(Number of Half-Lives) = 0.125

To find the Number of Half-Lives, we can take the logarithm base 1/2 of both sides:

log(1/2)^(Number of Half-Lives) = log(0.125)

Number of Half-Lives * log(1/2) = log(0.125)

Number of Half-Lives = log(0.125) / log(1/2)

Using a calculator, we can evaluate this expression:

Number of Half-Lives ≈ 3.0

So, it will take approximately 3 half-lives for the 18.0 g sample of iodine-131 to decay to a total of 2.25 g.

Finally, to determine the total time it takes, we multiply the number of half-lives by the half-life of iodine-131:

Total Time = Number of Half-Lives * Half-Life

Total Time = 3 * 8.07 days

Total Time ≈ 24.21 days

Therefore, it will take approximately 24.21 days for the 18.0 g sample of iodine-131 to decay to a total of 2.25 g.